Symmetric tensors

using TensorDec

We consider symmetric tensors or equivalently homogeneous polynomials, in the following variables:

using TensorDec, DynamicPolynomials
X = @polyvar x0 x1 x2;

A symmetric tensor of order d=4 and of rank 3.

d=4; F = (x0+x1+0.75x2)^d + 1.5*(x0-x1)^d -2.0*(x0-x2)^d

$ -1.68359375x2^{4} + 1.6875x1x2^{3} + 3.375x1^{2}x2^{2} + 3.0x1^{3}x2 + 2.5x1^{4} + 9.6875x0x2^{3} + 6.75x0x1x2^{2} + 9.0x0x1^{2}x2 - 2.0x0x1^{3} - 8.625x0^{2}x2^{2} + 9.0x0^{2}x1x2 + 15.0x0^{2}x1^{2} + 11.0x0^{3}x2 - 2.0x0^{3}x1 + 0.5x0^{4} $

The graph of the homogeneous polynomial $(x_0+x_1+0.75x_2)^4 + 1.5(x_0-x_1)^4 -2(x_0-x_2)^4$ in polar coordinates on the sphere looks like this:

We associate to $t$, the following (truncated) series in the dual variables, after substituting $x_0$ by 1:

Computing its decomposition

w, Xi = decompose(F);

yields the weights w

w

3-element Vector{Float64}:
 -8.000000000000021
  6.000000000001599
  6.566406250000015

and the corresponding points $\Xi$, which are the coefficient vectors of $x_0, x_1, x_2$ in the linear forms of the decomposition of the tensor F. They are normalized to have norm 1:

Xi

3×3 Matrix{Float64}:
 -0.707107      0.707107     0.624695
  4.71018e-19  -0.707107     0.624695
  0.707107      2.43289e-17  0.468521